Axiom:Axiom of Foundation

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For all non-empty sets, there is an element of the set that shares no element with the set.

That is:

$\forall S: \left({ \left({\exists x: x \in S}\right) \implies \exists y \in S: \forall z \in S: \neg \left({z \in y}\right) }\right)$

The antecedent states that $S$ is not empty.

It can also be stated as:

  • For every non-empty set $S$, there exists an element $x \in S$ such that $x$ and $S$ are disjoint.
  • A set contains no infinitely descending (membership) sequence.
  • A set contains a (membership) minimal element.

Also known as

The axiom of foundation is also known as the axiom of regularity.